How Long To Meet Driving Philadelphia To Pittsburgh Problem Solution

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In this article, we'll solve a classic problem involving distance, rate, and time. This is a common type of problem that appears in various contexts, from standardized tests to real-life scenarios. Understanding how to approach these problems can significantly enhance your problem-solving skills. Let's dive into the problem and break it down step by step.

Understanding the Problem

First, let’s clearly define the problem. Rich is driving from Philadelphia to Pittsburgh at a speed of 70 mph, while Michelle is driving from Pittsburgh to Philadelphia at 65 mph. The total distance between the two cities is 305 miles. The question we need to answer is: How long will it take for Rich and Michelle to meet? This is a classic relative motion problem, and to solve it, we need to consider how their speeds combine as they move towards each other.

Key Concepts

To solve this problem effectively, we need to understand a few key concepts:

  1. Distance, Rate, and Time Relationship: The fundamental relationship is Distance = Rate × Time. This formula is the cornerstone of solving motion problems. We can rearrange it to find any of the variables if we know the other two. For instance, Time = Distance / Rate and Rate = Distance / Time.
  2. Relative Speed: When two objects move towards each other, their speeds add up. This combined speed is known as the relative speed. It represents how quickly the distance between them is decreasing. In this case, since Rich and Michelle are driving towards each other, their relative speed is the sum of their individual speeds.
  3. Meeting Point: The point where Rich and Michelle meet is crucial. At the meeting point, the sum of the distances they have traveled will equal the total distance between the two cities. This understanding is key to setting up the equation to solve the problem.

Setting Up the Solution

With these concepts in mind, we can set up a solution. Let's denote the time it takes for them to meet as t (in hours). In time t, Rich will cover a distance of 70t miles, and Michelle will cover a distance of 65t miles. When they meet, the sum of these distances will be equal to the total distance between Philadelphia and Pittsburgh, which is 305 miles. This gives us the equation:

70t + 65t = 305

This equation represents the core of the problem. It states that the combined distance covered by Rich and Michelle is equal to the total distance between the cities. By solving this equation, we can find the time t it takes for them to meet.

Solving the Problem

Now that we have set up the equation, let's solve it step by step. This involves combining like terms, isolating the variable, and performing the necessary arithmetic operations.

Step-by-Step Solution

  1. Combine Like Terms: The equation we have is 70t + 65t = 305. Both terms on the left side have the variable t, so we can combine them: (70 + 65)t = 305. This simplifies to 135t = 305.
  2. Isolate the Variable: To find t, we need to isolate it on one side of the equation. We can do this by dividing both sides of the equation by 135: t = 305 / 135.
  3. Perform the Division: Now, we perform the division to find the value of t: t ≈ 2.259 hours. This is the time it takes for Rich and Michelle to meet, but it’s in decimal form. To make it more understandable, we can convert the decimal part into minutes.
  4. Convert to Hours and Minutes: The whole number part, 2, represents 2 hours. The decimal part, 0.259, represents a fraction of an hour. To convert this to minutes, we multiply it by 60 (since there are 60 minutes in an hour): 0.259 × 60 ≈ 15.54 minutes. So, the time it takes for them to meet is approximately 2 hours and 15.54 minutes.

Rounding to the Nearest Minute

In practical terms, we would typically round the time to the nearest minute. In this case, 15.54 minutes is very close to 16 minutes, so we can say that Rich and Michelle will meet in approximately 2 hours and 16 minutes.

Verification

To ensure our answer is correct, we can verify it by calculating the distances each person travels in this time and checking if their combined distances add up to 305 miles. In 2.259 hours:

  • Rich travels: 70 mph × 2.259 hours ≈ 158.13 miles
  • Michelle travels: 65 mph × 2.259 hours ≈ 146.835 miles

Adding these distances: 158.13 miles + 146.835 miles ≈ 304.965 miles, which is very close to 305 miles. The slight difference is due to rounding errors in our calculations. This verification step confirms that our solution is accurate.

Answers and Conclusion

The time it will take for Rich and Michelle to meet is approximately 2 hours and 15 minutes. This problem illustrates the application of the distance, rate, and time relationship, as well as the concept of relative speed. By understanding these principles, you can effectively solve similar problems in various contexts. Remember to break down the problem, identify the key concepts, set up the equation, and solve it step by step. Verification is also crucial to ensure the accuracy of your solution.

Practical Implications

This type of problem has practical implications in real-world scenarios, such as planning travel times, coordinating meetings, or calculating the speed and distance of moving objects. By mastering these problem-solving techniques, you can enhance your analytical skills and make informed decisions in various situations.

Common Mistakes to Avoid

When solving distance, rate, and time problems, there are several common mistakes to avoid:

  • Forgetting to Add Speeds: In relative motion problems where objects are moving towards each other, remember to add their speeds to find the relative speed.
  • Incorrectly Setting Up the Equation: Ensure that the equation accurately represents the relationship between distance, rate, and time. Double-check that the units are consistent (e.g., miles per hour and hours).
  • Rounding Errors: Be mindful of rounding errors, especially when dealing with decimals. Round at the end of the calculation to maintain accuracy.
  • Not Verifying the Solution: Always verify your solution to catch any errors. Check if the answer makes sense in the context of the problem.

By avoiding these common mistakes, you can improve your problem-solving accuracy and efficiency.

Additional Tips and Strategies

To further enhance your skills in solving distance, rate, and time problems, consider the following tips and strategies:

  • Draw Diagrams: Visual aids can help you understand the problem better. Draw a simple diagram to represent the distances, speeds, and directions of the objects involved.
  • Use Tables: Organize the information in a table with columns for distance, rate, and time. This can help you keep track of the given values and identify what you need to find.
  • Practice Regularly: The more you practice, the more comfortable you will become with these types of problems. Solve a variety of problems with different scenarios and complexities.
  • Break Down Complex Problems: If a problem seems overwhelming, break it down into smaller, more manageable parts. Solve each part separately and then combine the results.
  • Check Units: Always check that the units are consistent throughout the problem. If necessary, convert units to ensure they match (e.g., convert minutes to hours).

By incorporating these tips and strategies into your problem-solving approach, you can tackle even the most challenging distance, rate, and time problems with confidence.

Conclusion

In conclusion, solving distance, rate, and time problems like the one involving Rich and Michelle requires a clear understanding of the fundamental concepts, careful setup of the equation, and systematic problem-solving steps. By mastering these skills, you can effectively solve similar problems in various contexts and improve your overall analytical abilities. Remember to practice regularly, avoid common mistakes, and use helpful strategies to enhance your problem-solving proficiency. Whether you are preparing for an exam or simply seeking to improve your mathematical skills, understanding these concepts is invaluable.